Understanding Randomness

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Chapter 11

• Understanding Randomness

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Random Behavior

• We use the word random to mean unpredictable
  or undecipherable.
• But when we talk about randomness in statistics,
  its much more structured than that, especially
  in the long run.
• The most important assumption about randomness
  when used in statistics is that its fair.

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Random Behavior, contd

• Example If you tossed a coin once, the result is
  unpredictable (randoms usual meaning), but
  youre pretty confident that if you tossed the
  coin many times, youd get Heads about 50 of the
  time and Tails 50 of the time.
• This is called Random Behavior.
• What would you think if you tossed a coin many
  times and it came out 60 Heads? Would you
  conclude the coin was fair?

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Random Behavior, contd

• What about if it came out 70 Heads? Would you
  conclude the coin was fair?
• Statistics seeks to determine answers to these
  questions by investigating how probable it would
  be for Heads to come up 70 of the time under
  normal random behavior.
• If that probability were really small, wed say
  that getting Heads 70 of the time would not
  likely happen due to chance. Then wed conclude
  that the coin was not fair.

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Simulations

• Continuing with the coin toss example, we want to
  gain some insight into how many Heads ought to
  come up when a coin is tossed many times, and how
  common getting Heads 70 of the time might
  actually be.
• Simulation is a method used to model a real-world
  event like tossing a coin but without the time
  and expense that would usually be involved.

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Simulations, contd

• A simulation consists of a sequence of random
  outcomes that model a situation.
• The strength of any simulation lies in the fact
  that we can repeat a process a huge number of
  times. As the number of trials increases, the
  results become more reliable.
• Simulations not perfect! After all, simulation
  too is a model, much like our regression and
  normal models are.
• Simulations only help us conclude what could
  possibly happen, not what will happen.

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Simulations Important Terms
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Steps for a Simulation
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Steps for a Simulation, contd
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Another Example p. 268 34

• State the issue you are exploring. We want to
  know whether the likelihood of seeing 4 drivers
  out of 10 talking on cell phones while driving is
  high enough to call the legislators 12 rate
  into question.
• Identify the component(s) to be repeated. The
  component in this simulation is a driver of a
  single car in a series of cars passing by.
• Explain how youll model the outcome. We will
  generate a two-digit random number to represent
  each component using the TI-83 randInt function.
  A number from 0-11 will indicate a driver using a
  cell phone. All other numbers will indicate a
  driver not using a cell phone.

11
34, continued

• Explain how youll simulate the trial. A trial
  consists of 10 2-digit numbers, representing the
  drivers of the 10 cars. 100 trials will be
  generated. A 1 will be recorded if 4 or more
  numbers of the 10 are between 0 and 11
  otherwise, a 0 will be recorded, indicating
  that fewer than 4 drivers are talking on cell
  phones.
• State clearly what the response variable is. The
  response variable is how many of the 100 trials
  contained at least 4 numbers less than 12.
• Run several trials. After executing 100 trials,
  we will count how many trials contained 4 or more
  numbers lt 12 (i.e., cell phone drivers).

12
34, continued

• Analyze the response variable. Out of 100 groups
  of 10 two-digit numbers, ____ contained at least
  4 numbers between 0 and 11, making the
  probability of seeing 4 cell phone drivers out of
  10, _____.
• State your conclusion, in the context of the
  problem. Because the above probability, ____,
  is __lthigh, medium, lowgt_ compared to the 12
  cited by the legislator, we conclude that
  ____________________ _____________________________
  __________.